statements
1. $ \overleftrightarrow { c m } $ is a perpendicular bisector of $ \overline { a b } $ at point
m.
2. $ \overleftrightarrow { c m } \perp \overline { a b } $
3. $ \angle a m c $ and $ \angle b m c $ are right $ \angle $ s.
4.
5. $ m $ is the midpoint of $ \overline { a b } $
6. $ \overline { a m } \cong \overline { m b } $
7.
8. $ \triangle a m c \cong \triangle b m c $
9. $ \overline { a c } \cong \overline { b c } $
10. $ a c = b c $
a. $ \perp $ bisector
b. $ \angle a m c \cong \angle b m c $
c. midpoint
d. $ \overline { c m } \cong \overline { c m } $
e. def. of $ \perp $
f. $ c m = c m $
g. cpctc
h. $ \angle a m c \cong \angle a m c $

Question

Accepted Answer

# Explanation:
## Step1: Reason for statement 2
Since $ \overleftrightarrow{CM}$ is a perpendicular bisector (given), by the definition of perpendicular ($ \perp$), $ \overleftrightarrow{CM}\perp\overline{AB}$. So the reason for statement 2 is "Def. of $ \perp$" (option e).
## Step2: Reason for statement 3
If two lines are perpendicular, the angles formed are right angles. By the definition of perpendicular lines ($ \overleftrightarrow{CM}\perp\overline{AB}$), $ \angle AMC$ and $ \angle BMC$ are right angles. The reason is based on the "Def. of $ \perp$" (related to the formation of right - angles from perpendicular lines).
## Step3: Statement 4
Since $ \angle AMC$ and $ \angle BMC$ are right angles and we use the Right $ \angle$ Congruence Theorem (which states that all right angles are congruent),  the statement is $ \angle AMC\cong\angle BMC$ (option b).
## Step4: Reason for statement 6
Since $M$ is the mid - point of $ \overline{AB}$ (statement 5, by the definition of a mid - point ($M$ divides $ \overline{AB}$ into two equal parts), $ \overline{AM}\cong\overline{MB}$. The reason is "Def. of Midpoint" (option c).
## Step5: Statement 7
By the Reflexive Property of Congruence ($ \cong$), any segment is congruent to itself. So $ \overline{CM}\cong\overline{CM}$ (option d).
## Step6: Reason for statement 9
Since $ \triangle AMC\cong\triangle BMC$ (statement 8) by $SAS$ ( $ \overline{AM}\cong\overline{MB}$, $ \angle AMC\cong\angle BMC$, $ \overline{CM}\cong\overline{CM}$), then by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $ \overline{AC}\cong\overline{BC}$. The reason is "CPCTC" (option g).

# Answer:
2. e. Def. of $ \perp$
3. e. Def. of $ \perp$
4. b. $ \angle AMC\cong\angle BMC$
6. c. Midpoint
7. d. $ \overline{CM}\cong\overline{CM}$
9. g. CPCTC

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QUESTION IMAGE

Question

Explanation:

Step1: Reason for statement 2

Step2: Reason for statement 3

Step3: Statement 4

Step4: Reason for statement 6

Step5: Statement 7

Step6: Reason for statement 9

Answer: